Solitary and compacton solutions of fractional KdV-like equations

Abstract

In this paper, based on Jumarie's modified Riemann-Liouville derivative, we apply the fractional variational iteration method using He's polynomials to obtain solitary and compacton solutions of fractional KdVlike equations. The results show that the proposed method provides a very effective and reliable tool for solving fractional KdV-like equations, and the method can also be extended to many other fractional partial differential equations.

Closed-form travelling wave solutions to the nonlinear space-time fractional coupled Burgers’ equation

“…The closed-form travelling wave solutions of the fractional equations (Seadawy, 2017e;Baleanu, Diethelm, Scalas, & Trujillo, 2012;Islam, Akbar, & Azad, 2018;Liu, Li, Zhang, & Liu, 2015) are very helpful for understanding the mechanisms of the phenomena, as well as their further application in practical life. Numerous influential methods have been proposed to investigate the exact travelling wave solutions to nonlinear partial differential equations (NPDEs) of fractional order as well as integer order, as for instance the symmetry group method (El-Shiekh, 2018), the expðÀUðnÞÞ-expansion method (Kaplan & Akbulut, 2018), the direct algebraic method (Seadawy, 2012), the fractional sub-equation method (Alzaidy, 2013;Mohyud-Din, Nawaz, Azhar, & Akbar, 2017;Zhang & Zhang, 2011), the extended direct algebraic method (Seadawy, 2014;Seadawy, 2016a;Seadawy, 2016b), the Adomian decomposition method (El-Sayed, Behiry, & Raslan, 2010;Hu and He, 2016), the variational iteration method (Singh and Kumar, 2017;Tang, Fan, Zhao, & Wang, 2016), the modified extended direct algebraic mapping (Seadawy, 2016c), the auxiliary equation mapping method and direct algebraic mapping method (Seadawy & Lu 2016), the ðG 0 =GÞ-expansion method and its various modifications (Alam & Akbar, 2014;Feng, Li, & Wan, 2011;Islam, Akbar, & Azad, 2017), the amplitude ansatz method (Seadawy & Lu 2017;Seadawy, 2017a), multiple scales methods (Seadawy, 2017b), the homotopy perturbation method (Cherif, Belghaba, & Ziane, 2016;He, 1999), the extended auxiliary equation method (Seadawy, 2017c), the mathematical methods (Seadawy, 2017d). the differential transformation method (Sepasgozar, Faraji, & Valipour, 2017), the extended modified mapping met...…”